A New Geometry for Schools. By S. Barnard and J.
M. Child. (Macmillan and Co. 4s. 6d.)—This very useful book on elementary plane geometry embodies the views of the Mathe- matical Association and of the Cambridge Syndicate (appointed to consider the necessary changes in Mathematical Pass Examina- tions in the University) on the teaching of geometry. It com- prises a preliminary section on fundamental concepts, a practical section containing (inter die) practical constructions, and a theoretical section including the course of theorems recommended by the Syndicate as covering the whole of the first six books of Euclid. The method of teaching Euclid has of recent years greatly changed, and the system of introducing the subject by practical constructions is a change that has a great deal to be said for it, though it has some drawbacks. One of the great values of the old Euclid was the very fact of its lengthy verbal logical sequences. Prom the mathematical point of view this has, rightly, been disliked ; but from the point of view of logical training it was very valuable, and the loss of this first introduction to logic is considerable. Moreover, practical con- structions are perhaps likely to have a stunting effect on pure geometrical thought. This bcok has a very unfamiliar look, and its introduction to schools would have been facilitated, perhaps, by a closer association of the theorems with their Euclidean originals However, if we are to part with our old friend, perhaps it will be as well to tell the new generation nothing about him. The new geometry ought, we think, to have eliminated as far as possible Euclid's errors. It is a pity that the vicious method of super- position—for which nothing good can in theory be said—should be employed. The equality of angles in the Fourth Proposition of the first book should be treated as axiomatic. We, moreover, doubt if the authors of this book have really set out all the axioms they use. However, these are small points, and we hope that this careful and valuable book will be very widely used. He different parts make it suitable for scholars of all ages and all schools.